How would you explain the natural numbers to an alien devoid of a number instinct? You could try Peano arithmetic...

In the 1930s the logician Kurt Gödel showed that if you set out proper rules for mathematics, you lose the ability to decide whether certain statements are true or false. This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite contrived. But are they about to enter mainstream mathematics?

What's the nature of infinity? Are all infinities the same? And what happens if you've got infinitely many infinities? In this article **Richard Elwes** explores how these questions brought triumph to one man and ruin to another, ventures to the limits of mathematics and finds that, with infinity, you're spoilt for choice.

For millennia, puzzles and paradoxes have forced mathematicians to continually rethink their ideas of what proofs actually are. **Jon Walthoe** explains the tricks involved and how great thinkers like Pythagoras, Newton and Gödel tackled the problems.

Starting in this issue, PASS Maths is pleased to present a series of articles about proof and logical reasoning. In this article we give a brief introduction to deductive reasoning and take a look at one of the earliest known examples of mathematical proof.